Problem 40 - Calculus for Biology and Medicine

Question

Let $$ f(x)=\frac{2}{3+x}, \quad x>-3 $$ (a) Use a graphing calculator to graph \(f(x)\). (b) Find the range of \(f(x)\). (c) For which values of \(x\) is \(f(x)=1\) ? (d) Based on the graph in (a), explain in words why, for any value in the range of \(f(x)\), you can find exactly one value \(x \geq 0\) such that \(f(x)=a\). Determine \(x\) for general \(a\) by solving \(f(x)=a\).

Step-by-Step Solution

Answer

(a) Graph shows a hyperbola. (b) Range: \(y \neq 0\). (c) \(x = -1\) for \(f(x) = 1\). (d) One-to-one property; solve: \(x = \frac{2-3a}{a}\).

Step 1: Graph the function

To graph the function \(f(x) = \frac{2}{3+x}\), input it into a graphing calculator. You will observe a hyperbola that approaches the vertical line \(x = -3\) (vertical asymptote), and the horizontal line \(y = 0\) (horizontal asymptote) as \(x\) moves towards negative and positive infinity.

Step 2: Determine the range of the function

Since \(f(x) = \frac{2}{3+x}\), and the denominator \(3+x\) approaches positive and negative values, the function outputs all real numbers except zero. Therefore, the range of \(f(x)\) is \(y eq 0\).

Step 3: Solve for x when f(x) = 1

Set \(f(x) = 1\) and solve for \(x\): \[\frac{2}{3+x} = 1\] Multiply both sides by \(3+x\) to get: \[2 = 3 + x\] Subtract 3 from both sides: \[x = -1\] Thus, \(f(x) = 1\) when \(x = -1\).

Step 4: Explain the uniqueness of x for each value in the range

For any value \(a\) in the range of \(f(x)\), \(a eq 0\), there exists exactly one value of \(x\) such that \(f(x) = a\). This is because the equation is a transformation of a hyperbola, which is continuous and one-to-one in each branch.

Step 5: Solve f(x) = a for x

To find \(x\) when \(f(x) = a\), solve \[\frac{2}{3+x} = a\]. Multiply both sides by \(3+x\) to get: \[2 = a(3+x)\]. This simplifies to: \[2 = 3a + ax\]. Rearrange to solve for \(x\): \[ax = 2 - 3a\]. Finally, \[x = \frac{2 - 3a}{a}, \ a eq 0\].

Key Concepts

Function Graphing

Understanding how to graph rational functions, like \(f(x) = \frac{2}{3 + x}\), is essential. These types of functions often have very distinct characteristics due to the division of polynomials.
To graph, it's best to use a graphing calculator or software. Start by inputting the function into the calculator. You'll usually see a hyperbola, a peculiar shape for rational functions.
Pay attention to how the graph behaves near certain points. With \(f(x) = \frac{2}{3+x}\), the graph seems to stretch outwards towards the vertical line \(x = -3\), called the vertical asymptote. As you look further into the positive and negative ends, you may notice the graph nearing the horizontal line \(y = 0\), known as a horizontal asymptote.
These seemingly invisible lines help us understand how the function will behave towards more extensive positive and negative values of \(x\). They guide us, showing that the function will closely hug these lines but never touch them.

Range of Functions

The range of a function tells us all of the possible output values it can produce. For \(f(x) = \frac{2}{3+x}\), we observe that as \(x\) changes, the output or \(y\)-value also varies.
To determine the range of this particular function, examine the behavior of the graph. Notice that the function approaches, but never reaches, \(0\). This suggests \(y = 0\) isn't part of the range.
Thus, all real numbers except \(0\) can be the output or range of \(f(x)\). You can express this as \(y eq 0\). Getting comfortable with ranges means recognizing that sometimes not every \(y\)-value is possible.

Solving Equations

Equations involving rational functions can seem tricky, but they become manageable with practice. Suppose you want to know when \(f(x) = 1\) for our function \(f(x) = \frac{2}{3+x}\).
Start by setting the function equal to 1: \(\frac{2}{3+x} = 1\). Multiply both sides by the denominator, \(3+x\), to clear the fraction: \(2 = 3 + x\). Subtract 3 from both sides to isolate \(x\): \(x = -1\). This tells us that when \(x = -1\), the output \(y\) is 1.
Mastering this step gives you a solid foundation in solving similar equations, making once challenging problems become second nature.

Asymptotes

Asymptotes are the invisible lines that a graph approaches infinitely close to but never actually touches. Understanding asymptotes is crucial in graphing rational functions.
In the function \(f(x) = \frac{2}{3+x}\), the vertical asymptote occurs where \(x\) makes the denominator zero — specifically, \(x = -3\). This means as \(x\) gets closer to -3, the graph will skyrocket towards infinity or dip negatively, never crossing or touching the vertical line at \(x = -3\).
The horizontal asymptote is all about long-term behavior. For \(f(x) = \frac{2}{3+x}\), as \(x\) goes towards positive or negative infinity, the \(y\)-value nears \(0\). Thinking of asymptotes like this helps predict the curve's behavior, characterizing the spread of the graph across the plane. Recognizing these lines ensures you can graph practically any rational function confidently, making asymptotic behavior a fundamental concept.